Single Degree of Freedom System Free Vibration

Transcription

1 Iegriy, Professioalism, & Erepreership Maa Kliah : Diamika Srkr & Pegaar Rekayasa Kegempaa Kode : CIV 308 SKS : 3 SKS Sigle Degree of Freedom Sysem Free Vibraio Perema -

2 Iegriy, Professioalism, & Erepreership TIU : Mahasisa dapa mejelaska eag eori diamika srkr. Mahasisa dapa memba model maemaik dari masalah ekis yag ada sera mecari solsiya. TIK : Mahasisa dapa memformlasika persamaa gerak sisem srkr berderaja kebebasa ggal yag bergear bebas

3 Iegriy, Professioalism, & Erepreership Sb Pokok Bahasa : Geara Bebas SDoF Tak Teredam Frekesi da Periode Geara Bebas SDoF Dega Redama

4 Iegriy, Professioalism, & Erepreership Udamped SDoF Free-Vibraio A srcre is said o be dergoig free vibraio he () i is disrbed from is saic eqilibrim posiio ad he k alloed o vibrae iho ay exeral dyamic exciaio. m The differeial eqaio goverig free vibraio of he sysems iho dampig (c = 0 & p() = 0) is : m k 0 (1) This eqaio called secod order homogeeos differeial eqaio

5 Iegriy, Professioalism, & Erepreership To fid he solio of Eq. (1), assmig a rial solio give by : Acos (.a) B si (.b) Where A ad B are cosas depedig o he iiiaio of he moio hile is a qaiy deoig a physical characerisic of he sysem.

6 Iegriy, Professioalism, & Erepreership The sbsiio eq.(.a&b) io eq. (1) gives : m k Acos 0 (3) If eq.(3) is o be saisfied a ay ime, he facor i pareheses ms be eqal o zero, or : k m k (4) m is ko as he aral circlar freqecy of he sysem (rad/s)

7 Iegriy, Professioalism, & Erepreership Sice eiher eq.(.a) or (.b) is a solio of eq.(1), ad sice he differeial eqaio is liear, he sperposiio of hese o solio is also solio : Acos B si (5) The expressio for velociy, ú, is : A si B cos Cosa A ad B, ill be deermied based o iiial codiio (a = 0) : 0 0 (6)

8 Iegriy, Professioalism, & Erepreership Sbjec o hese iiial codiios, he solio o he eq. (5) ill be : 0 cos 0 si (7) 0 0 0

9 Iegriy, Professioalism, & Erepreership Freqecy ad Period The ime reqired for he damped sysem o complee oe cycle of free vibraio is he aral period of vibraio, T (i secod) T A sysem execes 1/T cycles i 1 sec. This aral cyclic freqecy of vibraio (i Hz) is deoed by : (8) f (9)

10 Iegriy, Professioalism, & Erepreership Exercise Deermie aral freqecy ad aral period from each sysem belo W =,5 os E, I, L W k EI 40x40 cm 3 m k (a) E= MPa (b)

11 Iegriy, Professioalism, & Erepreership Assigme 1 If sysem (b) have iiial codiio 0 3 cm & 0 0 cm/s a) Deermie he displaceme a = sec b) Plo a ime hisory of displaceme respose W =,5 os of he sysem, for = 0 s il = 5 s. EI 40x40 cm 3 m E= MPa (b)

12 Iegriy, Professioalism, & Erepreership Damped SDoF Free-Vibraio If dampig is prese i he sysem, he differeial eqaio goverig free vibraio of he sysems is : m c k 0 The rial solio o saisfy Eq. (14) is (10) p Ce (11) Sbie Eq. (11) o Eq. (10) yield he characerisic eq. : mp cp k 0 (1) c k m ()

13 Iegriy, Professioalism, & Erepreership The roos of he qadraic Eq. (1) are : p 1, c m c m k m (13) Ths he geeral solio of Eq. (10) is : p1 p C1e Ce (14) Three disic cases may occr depeds o he sig der he radical i Eq. (13)

14 Iegriy, Professioalism, & Erepreership 1. Criically Damped Sysem For criical dampig : c c m km (15) cr The expressio der he radical i Eq. (13) is eqal o zero, ad he roos are eqal, hey are : p 1 p ccr m The geeral solio of Eq. (14) is : A Be (16) (17)

15 Iegriy, Professioalism, & Erepreership 1. Criically Damped Sysem Cosa A ad B, ill be deermied based o iiial codiio (a = 0) : 0 0 Sbsie he iiial codiio, yield : e (18)

16 Iegriy, Professioalism, & Erepreership. Overdamped Sysem I a overdamped sysem, he dampig coefficie is greaer ha he vale for criical dampig (c > c cr ) EoM of a overdamped sysem de o iiial codiio is : Where : ξ is he dampig raio (%) D D Be Ae e D D B A r D c c 1 (19)

17 Iegriy, Professioalism, & Erepreership 3. Uderdamped Sysem Whe he vale of he dampig coefficie is less ha he criical vale (c < c cr ), he roos of he characerisic eq. (1) are complex cojgaes, so ha : c k c p 1, i i 1 m m m Regardig he Eler s eqaios hich relae expoeial ad rigoomeric fcios : e ix cos x i si x e ix cos x i si x

18 Iegriy, Professioalism, & Erepreership 3. Uderdamped Sysem The sbsiio of he roos p 1 ad p io Eq. (14) ogeher ih he se of Eler eqaio, gives : e Acos B si (0) D D Where 1 Usig iiial codiio, D e 0cosD 0 D si D (1)

19 Iegriy, Professioalism, & Erepreership May ypes of srcres, sch as bildigs, bridges, dams, clear poer plas, offshore srcres, ec., all fall io derdamped sysem (c < c cr ), becase ypically heir dampig raio is less ha 0,10.

20 Iegriy, Professioalism, & Erepreership Logarihmic Decreme The raio i / i+1 of sccessive peaks (maxima) is : i exp i1 1 () The aral logarihm of his raio, called he logarihmic decreme, d l i i1 1 (3) If z is small, his gives a approximae eqaio : d z

21 Iegriy, Professioalism, & Erepreership Over j cycles he moio decreases from 1 o j+1. This raio is give by : Therefore, j1 3 4 j1 3 j e jd (4) 1 1 d l j j 1 (5)

22 Iegriy, Professioalism, & Erepreership Exercise Weigh, W = mg A oe-sory bildig is idealized as a rigid girder sppored by eighless colms. I order o evalae he dyamic properies of his srcre, a free-vibraio es is made, i hich he roof c sysem (rigid girder) is displaced laerally by a hydralic jack k/ ad he sddely k/ released. Drig he jackig operaio, i is observed ha a force of 10 kg is reqired o displace he girder cm. Afer he isaaeos release of his iiial displaceme, Fid : he maximm displaceme o he firs rer sig is oly cm W ad he period of his displaceme cycle is T = 1.40 sec. f ad d,, c, D 6

23 Iegriy, Professioalism, & Erepreership Assigme If sysem i he figre have iiial codiio Plo a ime hisory of displaceme respose of he sysem, for = 0 s il = 5 s, if : W =,5 os a. ξ = 5% b. ξ = 100% c. ξ = 15% 0 3 cm & 0 0 cm/s EI 40x40 cm 3 m E= MPa (b)